Questions S1 (1967 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
CAIE S1 2018 June Q5
5 A game is played with 3 coins, \(A , B\) and \(C\). Coins \(A\) and \(B\) are biased so that the probability of obtaining a head is 0.4 for coin \(A\) and 0.75 for coin \(B\). Coin \(C\) is not biased. The 3 coins are thrown once.
  1. Draw up the probability distribution table for the number of heads obtained.
  2. Hence calculate the mean and variance of the number of heads obtained.
CAIE S1 2018 June Q6
6 The diameters of apples in an orchard have a normal distribution with mean 5.7 cm and standard deviation 0.8 cm . Apples with diameters between 4.1 cm and 5 cm can be used as toffee apples.
  1. Find the probability that an apple selected at random can be used as a toffee apple.
  2. 250 apples are chosen at random. Use a suitable approximation to find the probability that fewer than 50 can be used as toffee apples.
CAIE S1 2018 June Q7
7 Find the number of ways the 9 letters of the word SEVENTEEN can be arranged in each of the following cases.
  1. One of the letter Es is in the centre with 4 letters on either side.
  2. No E is next to another E.
    5 letters are chosen from the 9 letters of the word SEVENTEEN.
  3. Find the number of possible selections which contain exactly 2 Es and exactly 2 Ns.
  4. Find the number of possible selections which contain at least 2 Es.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE S1 2019 June Q1
1 The times, \(t\) seconds, taken to swim 100 m were recorded for a group of 9 swimmers and were found to be as follows. $$\begin{array} { l l l l l l l l l } 95 & 126 & 117 & 135 & 120 & 125 & 114 & 119 & 136 \end{array}$$
  1. Find the values of \(\Sigma ( t - 120 )\) and \(\Sigma ( t - 120 ) ^ { 2 }\).
  2. Using your values found in part (i), calculate the variance of \(t\).
CAIE S1 2019 June Q2
3 marks
2 Jameel has 5 plums and 3 apricots in a box. Rosa has \(x\) plums and 6 apricots in a box. One fruit is chosen at random from Jameel's box and one fruit is chosen at random from Rosa's box. The probability that both fruits chosen are plums is \(\frac { 1 } { 4 }\). Write down an equation in \(x\) and hence find \(x\). [3]
CAIE S1 2019 June Q3
3 A fair six-sided die is thrown twice and the scores are noted. Event \(X\) is defined as 'The total of the two scores is 4'. Event \(Y\) is defined as 'The first score is 2 or 5'. Are events \(X\) and \(Y\) independent? Justify your answer.
CAIE S1 2019 June Q4
4 The Mathematics and English A-level marks of 1400 pupils all taking the same examinations are shown in the cumulative frequency graphs below. Both examinations are marked out of 100 .
\includegraphics[max width=\textwidth, alt={}, center]{be6c6525-a20c-42d0-8fef-1cd254baaa76-06_1682_1246_404_445} Use suitable data from these graphs to compare the central tendency and spread of the marks in Mathematics and English.
CAIE S1 2019 June Q5
5 In a certain country the probability that a child owns a bicycle is 0.65 .
  1. A random sample of 15 children from this country is chosen. Find the probability that more than 12 own a bicycle.
  2. A random sample of 250 children from this country is chosen. Use a suitable approximation to find the probability that fewer than 179 own a bicycle.
CAIE S1 2019 June Q6
6 At a funfair, Amy pays \(
) 1$ for two attempts to make a bell ring by shooting at it with a water pistol.
  • If she makes the bell ring on her first attempt, she receives \(
    ) 3\( and stops playing. This means that overall she has gained \)\\( 2\).
  • If she makes the bell ring on her second attempt, she receives \(
    ) 1.50\( and stops playing. This means that overall she has gained \)\\( 0.50\).
  • If she does not make the bell ring in the two attempts, she has lost her original \(
    ) 1$.
The probability that Amy makes the bell ring on any attempt is 0.2 , independently of other attempts.
  1. Show that the probability that Amy loses her original \(
    ) 1$ is 0.64 .
  2. Complete the probability distribution table for the amount that Amy gains.
    Amy's gain (\$)
    Probability0.64
  3. Calculate Amy's expected gain.
CAIE S1 2019 June Q7
7 The weight of adult female giraffes has a normal distribution with mean 830 kg and standard deviation 120 kg .
  1. There are 430 adult female giraffes in a particular game reserve. Find the number of these adult female giraffes which can be expected to weigh less than 700 kg .
  2. Given that \(90 \%\) of adult female giraffes weigh between \(( 830 - w ) \mathrm { kg }\) and \(( 830 + w ) \mathrm { kg }\), find the value of \(w\).
    The weight of adult male giraffes has a normal distribution with mean 1190 kg and standard deviation \(\sigma \mathrm { kg }\).
  3. Given that \(83.4 \%\) of adult male giraffes weigh more than 950 kg , find the value of \(\sigma\).
CAIE S1 2019 June Q8
8 Freddie has 6 toy cars and 3 toy buses, all different. He chooses 4 toys to take on holiday with him.
  1. In how many different ways can Freddie choose 4 toys?
  2. How many of these choices will include both his favourite car and his favourite bus?
    Freddie arranges these 9 toys in a line.
  3. Find the number of possible arrangements if the buses are all next to each other.
  4. Find the number of possible arrangements if there is a car at each end of the line and no buses are next to each other.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE S1 2019 June Q1
1 Two ordinary fair dice are thrown and the numbers obtained are noted. Event \(S\) is 'The sum of the numbers is even'. Event \(T\) is 'The sum of the numbers is either less than 6 or a multiple of 4 or both'. Showing your working, determine whether the events \(S\) and \(T\) are independent.
CAIE S1 2019 June Q2
2 The volume of ink in a certain type of ink cartridge has a normal distribution with mean 30 ml and standard deviation 1.5 ml . People in an office use a total of 8 cartridges of this ink per month. Find the expected number of cartridges per month that contain less than 28.9 ml of this ink.
CAIE S1 2019 June Q3
3 The probability that Janice will buy an item online in any week is 0.35 . Janice does not buy more than one item online in any week.
  1. Find the probability that, in a 10 -week period, Janice buys at most 7 items online.
  2. The probability that Janice buys at least one item online in a period of \(n\) weeks is greater than 0.99 . Find the smallest possible value of \(n\).
CAIE S1 2019 June Q4
4 It is known that 20\% of male giant pandas in a certain area weigh more than 121 kg and \(71.9 \%\) weigh more than 102 kg . Weights of male giant pandas in this area have a normal distribution. Find the mean and standard deviation of the weights of male giant pandas in this area.
CAIE S1 2019 June Q5
5 Maryam has 7 sweets in a tin; 6 are toffees and 1 is a chocolate. She chooses one sweet at random and takes it out. Her friend adds 3 chocolates to the tin. Then Maryam takes another sweet at random out of the tin.
  1. Draw a fully labelled tree diagram to illustrate this situation.
  2. Draw up the probability distribution table for the number of toffees taken.
  3. Find the mean number of toffees taken.
  4. Find the probability that the first sweet taken is a chocolate, given that the second sweet taken is a toffee.
CAIE S1 2019 June Q6
6
  1. Give one advantage and one disadvantage of using a box-and-whisker plot to represent a set of data.
  2. The times in minutes taken to run a marathon were recorded for a group of 13 marathon runners and were found to be as follows. $$\begin{array} { l l l l l l l l l l l l l } 180 & 275 & 235 & 242 & 311 & 194 & 246 & 229 & 238 & 768 & 332 & 227 & 228 \end{array}$$ State which of the mean, mode or median is most suitable as a measure of central tendency for these times. Explain why the other measures are less suitable.
  3. Another group of 33 people ran the same marathon and their times in minutes were as follows.
    190203215246249253255254258260261
    263267269274276280288283287294300
    307318327331336345351353360368375
    (a) On the grid below, draw a box-and-whisker plot to illustrate the times for these 33 people.
    \includegraphics[max width=\textwidth, alt={}, center]{f4d040a2-6a04-49ce-98ac-8ba5c515f905-09_611_1202_1270_555}
    (b) Find the interquartile range of these times.
CAIE S1 2019 June Q7
7
  1. A group of 6 teenagers go boating. There are three boats available. One boat has room for 3 people, one has room for 2 people and one has room for 1 person. Find the number of different ways the group of 6 teenagers can be divided between the three boats.
  2. Find the number of different 7-digit numbers which can be formed from the seven digits 2, 2, 3, 7, 7, 7, 8 in each of the following cases.
    1. The odd digits are together and the even digits are together.
    2. The 2 s are not together.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE S1 2019 June Q1
1 The time taken, in minutes, by a ferry to cross a lake has a normal distribution with mean 85 and standard deviation 6.8.
  1. Find the probability that, on a randomly chosen occasion, the time taken by the ferry to cross the lake is between 79 and 91 minutes.
  2. Over a long period it is found that \(96 \%\) of ferry crossings take longer than a certain time \(t\) minutes. Find the value of \(t\).
CAIE S1 2019 June Q2
2 Megan sends messages to her friends in one of 3 different ways: text, email or social media. For each message, the probability that she uses text is 0.3 and the probability that she uses email is 0.2 . She receives an immediate reply from a text message with probability 0.4 , from an email with probability 0.15 and from social media with probability 0.6 .
  1. Draw a fully labelled tree diagram to represent this information.
  2. Given that Megan does not receive an immediate reply to a message, find the probability that the message was an email.
CAIE S1 2019 June Q3
3 Mr and Mrs Keene and their 5 children all go to watch a football match, together with their friends Mr and Mrs Uzuma and their 2 children. Find the number of ways in which all 11 people can line up at the entrance in each of the following cases.
  1. Mr Keene stands at one end of the line and Mr Uzuma stands at the other end.
  2. The 5 Keene children all stand together and the Uzuma children both stand together.
CAIE S1 2019 June Q4
4
  1. Find the number of ways a committee of 6 people can be chosen from 8 men and 4 women if there must be at least twice as many men as there are women on the committee.
  2. Find the number of ways a committee of 6 people can be chosen from 8 men and 4 women if 2 particular men refuse to be on the committee together.
CAIE S1 2019 June Q5
5 On average, \(34 \%\) of the people who go to a particular theatre are men.
  1. A random sample of 14 people who go to the theatre is chosen. Find the probability that at most 2 people are men.
  2. Use an approximation to find the probability that, in a random sample of 600 people who go to the theatre, fewer than 190 are men.
CAIE S1 2019 June Q6
6 A fair five-sided spinner has sides numbered 1, 1, 1, 2, 3. A fair three-sided spinner has sides numbered \(1,2,3\). Both spinners are spun once and the score is the product of the numbers on the sides the spinners land on.
  1. Draw up the probability distribution table for the score.
    \includegraphics[max width=\textwidth, alt={}, center]{da4a61b9-f55d-40ed-a721-a6aee962f0d6-08_67_1569_484_328}
  2. Find the mean and the variance of the score.
  3. Find the probability that the score is greater than the mean score.
CAIE S1 2019 June Q7
7 The times in minutes taken by 13 pupils at each of two schools in a cross-country race are recorded in the table below.
Thaters School38434852545657585861626675
Whitefay Park School45475356566164666973757883
  1. Draw a back-to-back stem-and-leaf diagram to illustrate these times with Thaters School on the left.
  2. Find the interquartile range of the times for pupils at Thaters School.
    The times taken by pupils at Whitefay Park School are denoted by \(x\) minutes.
  3. Find the value of \(\Sigma ( x - 60 ) ^ { 2 }\).
  4. It is given that \(\Sigma ( x - 60 ) = 46\). Use this result, together with your answer to part (iii), to find the variance of \(x\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.